Re: Summary, was Re: Every time ..., was Re: General formula

[Ed Gerck <egerck@xxxxxxxxxx>]

Tony Bartoletti wrote:

> Ed,
> [snip]
>
>
> But perhaps the problem is that I cannot find REAL variables that
> behave this way.  As hard as I tried, I could only come up with
> examples that support your formula.

that is not bad ... to agree is also valid ;-)

> To wit:
>
>    Consider a very old stick of dynamite, so unstable that it
>    may self-detonate at any time, with an expected life of 1 year.
>
>    Now consider a box of 100 such sticks.  IN ISOLATION, each
>    stick has expected lifetime 1 year.  But given 100 such sticks,
>    the likelyhood that at least one stick will detonate within a
>    weeks time is rather large.  So, packaged together in the box,
>    the collection of dynamite sticks will likely not last a week.
>    (When one of them goes boom, it is almost a certainty that the
>    others will follow!)

I think this is a very illuminating example ;-)

Indeed, a certificate behaves this way -- after all, any attribute that is
in the cert....  is there because it is needed and the cert does "explode"
if any attribute "explodes". At least, it will "explode" as a whole, though
it may survive in "pieces".

Your other question (attribute redundancy) depends on the assumed
redudancy model. In the model  I suggested in my former reply to you,
the number of *effective* attributes is reduced as the inter-attribute
redundancy is increased -- so that when inter-attribute redundancy
is 100%, the number of attributes is only one and that is why 1/T =
1/To.

This redundancy model has some real-world significance. It models
a group that gradually loses individual identities by "quantum jumps"
into a collective identity, which is the final collective attribute. In other
words, as you increase redundancy, the number of individuals will
steadly decrease until only the collective is left. Thus, we need to take
into account also *that* attribute lifetime, the "collective attribute
lifetime". Which can have, interestingly enough, the same lifetime,
a lower lifetime or even a higher lifetime than each individual
attribute. Thus, the first formula is

  To = collective attribute lifetime = x years
  T1 = T2 = ... = T100 = individual lifetime = 1 year

and, to begin, suppose there is no collective yet:

   1/T = 1/T1 + ... + 1/T100 => T = 1/100 year

but, as redundancy is increased to the next notch, n individuals
are lost to the collective. Thus, the second formula has less attributes
to sum up and we obtain the partial expression:

  1/T = 1/To + 1/T1 + ... + 1/T(100 - n) => T =  x/((100-n)*x +1) years

and, as redundancy is further increased, there is a time when all
individuals are lost to the collective and n = 100, so that we only
have the collective to take into account:

  1/T = 1/To = x years

which is *also* the result we obtain by making n =100 in the partial
expression for n above:

 T =  x/((100-n)*x +1) years => x/((100-100)*x +1) = x years

Note thus that there is a progressive change from "1 year" to "x years",
going over  x/((100-n)*x +1) years, as redundancy is varied from 0%
to 100%. However, as I wrote above, one may have x >1, x < 1 or
x =1 -- as a function of the operational  context.


Is this what you had in mind?

Cheers,

Ed Gerck